Cardinality of product sets in torsion-free groups and applications in group algebras

TL;DR

This paper establishes lower bounds on the size of product sets in torsion-free groups with unique product properties and applies these results to group algebras, improving existing bounds in the literature.

Contribution

It provides new lower bounds for product set sizes in specific non-abelian groups and applies these to derive bounds in group algebra support sizes, extending previous results to arbitrary fields.

Findings

For non-abelian subsets, |BC| ≥ |B| + |C| + 2 when |B| ≥ 7.

For certain 3-element subsets, |BC| ≥ |B| + 5 when |B| ≥ 7.

If αβ=0 with |supp(α)|=3, then |supp(β)| ≥ 12.

Abstract

Let $G$ be a unique product group, i.e., for any two finite subsets $A$ and $B$ of $G$ there exists $x\in G$ which can be uniquely expressed as a product of an element of $A$ and an element of $B$. We prove that, if $C$ is a finite subset of $G$ containing the identity element such that $\langle C\rangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|\geq 7$, $|BC|\geq |B| + |C| + 2$. Also, we prove that if $C$ is a finite subset containing the identity element of a torsion-free group $G$ such that $|C| = 3$ and $\langle C\rangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|\geq 7$, $|BC|\geq |B| + 5$. Moreover, if $\langle C\rangle$ is not isomorphic to the Klein bottle group, i.e., the group with the presentation $\langle x, y \;|\; xyx = y\rangle$, then for all subsets $B$ of G with $|B|\geq 5$, $|BC|\geq |B| + 5$. The support of an element $\alpha =\sum_{x\in G} \alpha_x x$ a group algebra $F[G]$ ($F$ is any field), denoted by $supp(\alpha)$, is the set $\{x\in G \;|\; \alpha_x \neq 0\}$. By the latter result, we prove that if $\alpha \beta = 0$ for some non-zero $\alpha,\beta \in F[G]$ such that $|supp(\alpha)| = 3$, then $|supp(\beta)|\geq 12$. Also, we prove that if $\alpha \beta= 1$ for some $\alpha \beta \in F[G]$ such that $|supp(\alpha)| = 3$, then |supp(\alpha)|\geq 10$. These results improve a part of results in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667-693] and Dykema et al. [Exp. Math., 24 (2015), 326-338] to arbitrary fields, respectively.